A two-step symmetric method for charged-particle dynamics in a normal or strong magnetic field
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A two‑step symmetric method for charged‑particle dynamics in a normal or strong magnetic field Bin Wang1,2 · Xinyuan Wu3,4 · Yonglei Fang5 Received: 15 January 2019 / Revised: 31 October 2019 / Accepted: 14 August 2020 / Published online: 26 August 2020 © Istituto di Informatica e Telematica (IIT) 2020
Abstract The study of the long time conservation for numerical methods poses interesting and challenging questions from the point of view of geometric integration. In this paper, we analyze the long time energy and magnetic moment conservations of two-step symmetric methods for charged-particle dynamics. A two-step symmetric method is proposed and its long time behaviour is shown not only in a normal magnetic field but also in a strong magnetic field. The approaches to dealing with these two cases are based on the backward error analysis and modulated Fourier expansion, respectively. It is obtained from the analysis that the method has better long time conservations than the variational method which was researched recently in the literature. Keywords Charged-particle dynamics · Two-step symmetric methods · Backward error analysis · Modulated Fourier expansion Mathematics Subject Classification 65L05 · 65P10 · 78A35 · 78M25
* Bin Wang [email protected] Xinyuan Wu [email protected] Yonglei Fang [email protected] 1
School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an 710049, Shannxi, People’s Republic of China
2
Mathematisches Institut, University of Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany
3
Department of Mathematics, Nanjing University, Nanjing 210093, People’s Republic of China
4
School of Mathematical Sciences, Qufu Normal University, Qufu 273165, People’s Republic of China
5
School of Mathematics and Statistics, Zaozhuang University, Zaozhuang 277160, People’s Republic of China
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B. Wang et al.
1 Introduction The numerical investigation for charged-particle dynamics has received much attention in the last few decades (see e.g. [1, 8, 9, 12, 13, 17]). In this paper, we analyze the long time conservations of two two-step symmetric methods for solving charged-particle dynamics of the form (see [10])
1 ẍ = ẋ × B(x) + F(x), 𝜖
x(0) = x0 ,
x(0) ̇ = ẋ 0 , t ∈ [0, T],
(1)
where B(x) = ∇ × A(x) is a magnetic field with the vector potential A(x) ∈ ℝ3 , the position of a particle moving in this field is denoted by x(t) ∈ ℝ3 , and F(x) = −∇U(x) is an electric field with the scalar potential U(x). The energy of this dynamics
E(x, v) =
1 2 |v| + U(x) 2
(2)
is preserved exactly along the solution x and the velocity v = ẋ of the particle. It is assumed that the initial values are bounded as
x0 = O(1), v0 ∶= ẋ 0 = O(1).
(3)
In this work, we focus on the study of the following two regimes of 𝜖: – one regime is that 𝜖 in (1) is assumed to be one which means that the magnetic field is “normal”; – the other regime is that 𝜖 in (1) is assumed to satisfy 0 < 𝜖 ≪ 1 which means that the magnetic field is “strong”. For the
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